Let A = 1, 2, , 23 . Let R₁ = (x, y) A A : x^2 - y^2 is a multiple of 5 and R₂ = (x, y) A A : x - y is a multiple of 5 . The number of elements in the relation R₁ - R₂ is ____.

Step-by-step solution

The relation R₁ requires x^2 - y^2 0 5 , which implies (x - y)(x + y) 0 5 . Since 5 is a prime, this means either x y 5 or x -y 5 . The relation R₂ requires x y 5 . The set difference R₁ - R₂ consists of pairs (x, y) that belong to R₁ but not to R₂ . Thus, we need x -y 5 and x y 5 . If x -y 5 and x y 5 , adding the two congruences gives 2x 0 5 , which implies x 0 5 . Therefore, the condition x y 5 is equivalent to excluding the cases where x 0 5 . So, we need to count the pairs (x, y) such that x + y 0 5 and x 0 5